We discuss the algebra behind the matrix reduction algorithm for persistent homology, as presented in the paper ''Computing Persistent Homology'' by Afra Zomorodian and Gunnar Carlsson, in the lens of the more modern characterization of persistence modules as functors from a poset category to a category of vector spaces over a field adopted by authors such as Peter Bubenik, Frederik Chazal, and Ulrich Bauer.
We show that the interleaving distance between the persistent singular homology and the persistent Čech homology of a homologically locally connected filtration consisting of paracompact Hausdorff spaces is 0.
本文发展了加权持久同调理论。1990年，Robert J. Dawson首先深入研究了加权单纯复形的同调性。我们推广了加权单纯复形的定义和加权单纯复形的同调性，使权可以在整环$R$中存在。然后我们研究由此产生的加权持久同调。我们表明，加权持久同源性可以区分过滤，普通的持久同源性不区分。例如，如果有一个点被认为是特殊的，加权持久同调可以告诉包含该点的循环何时形成或消失。
Hypergraph is the most general model for complex networks involving group interactions. Taking the ideas of path homology from Alexander Grigor'yan, Yong Lin, Yuri Muranov and Shing-Tung Yau [18-22], Stephane Bressan, Jingyan Li and the authors of this article introduced embedded homology of hypergraphs [6] in 2019, which has leaded to successful applications in protein-ligand binding network [24, 25] in 2021. A fundamental question arising from practical applications is about the stability of the persistent embedded homology of hypergraphs. In this paper, we prove the stability of the persist…
Persistent magnitude homology on finite metric space
Chongqing University of Technology（重庆理工大学）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）
Computation of degree-1 persistent homology on larger point-clouds using the Reduced Vietoris-Rips filtration
Australian National University（澳大利亚国立大学）；Centre for Systems Biology Dresden（德累斯顿系统生物学中心）；Max Planck Institute of Cell Biology and Genetics（马克斯·普朗克细胞生物学与遗传学研究所）；Max Planck institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）
This paper introduces persistent homology, which is a powerful tool to characterize the shape of data using the mathematical concept of topology. We explain the fundamental idea of persistent homology from scratch using some examples. We also review some applications of persistent homology to materials researches and software for persistent homology data analysis. HomCloud, one of persistent homology software, is especially featured in this paper.
Persistent homology is a branch of computational algebraic topology that studies shapes and extracts features over multiple scales. In this paper, we present an unsupervised approach that uses persistent homology to study divergent behavior in agricultural point cloud data. More specifically, we build persistence diagrams from multidimensional point clouds, and use those diagrams as the basis to compare and contrast different subgroups of the population. We apply the framework to study the cold hardiness behavior of 5 leading grape cultivars, with real data from over 20 growing seasons. Our re…
Consider the space of continuous functions on a geometric tree $X$ whose persistent homology gives rise to a finite generic barcode $D$. We show that there are exactly as many path connected components in this space as there are merge trees whose barcode is $D$. We find that each component is homotopy equivalent to a configuration space on $X$ with specialized constraints encoded by the merge tree. For barcodes $D$ with either one or two intervals, our method also allows us to compute the homotopy type of this space of functions.
Graph-based representations of point-cloud data are widely used in data science and machine learning, including epsilon-graphs that contain edges between pairs of data points that are nearer than epsilon and kNN-graphs that connect each point to its k-nearest neighbors. Recently, topological data analysis has emerged as a family of mathematical and computational techniques to investigate topological features of data using simplicial complexes. These are a higher-order generalization of graphs and many techniques such as Vietoris-Rips (VR) filtrations are also parameterized by a distance epsilo…
Institute of Mathematics for Industry, Kyushu University（九州大学产业数学研究所）；School of Interdisciplinary Mathematical Sciences, Meiji University（明治大学跨学科数理科学学院）
We introduce Cubical Ripser for computing persistent homology of image and volume data (more precisely, weighted cubical complexes). To our best knowledge, Cubical Ripser is currently the fastest and the most memory-efficient program for computing persistent homology of weighted cubical complexes. We demonstrate our software with an example of image analysis in which persistent homology and convolutional neural networks are successfully combined. Our open-source implementation is available online.
Given a metric space $X$ and a subspace $A\subset X$, we prove $A$ can generate various algebraic elements in persistent homology of $X$. We call such elements (algebraic) footprints of $A$. Our results imply that footprints typically appear in dimensions above the dimension of $A$. Higher-dimensional persistent homology thus encodes lower-dimensional geometric features of $X$. We pay special attention to a specific type of geodesics in a geodesic surface $X$ called geodesic circles. We explain how they may generate non-trivial odd-dimensional and two-dimensional footprints. In particular, we …
Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations performed on many similar filtered topological spaces which is based on updating associated matrix factorizations. Our approach improves the update scheme of Cohen-Steiner, Edelsbrunner, and Morozov for permutations by additionally handling addition and deletion of cells in a filtered topological space and by processing changes in a single batch. We show that the c…
